VW - CVM 1.1

# The Rectangular Ingredients

The coordinates for this lattice are particularly simple: X and Y are simply proportional to x and y. The lattice waves are as in the general lattice , but here we find it convenient to reorganize the typical term as

anmcos(nX) cos(mY) + bnmcos(nX) sin(mY) + cnmsin(nX) cos(mY) + dnmsin(nX) sin(mY)

Among the isometries that will play a role in this analysis, h, d, v, and R are as before . Here, S denotes a reflection about a horizontal line through the center of the cell; Sv is the similar vertical reflections. Likewise, L and Lv are glide reflections with axes through the center of the rectangle. A and B are the mirror and glide parallel to M but a quarter of the way down the cell; when subscripted with v they are a quarter of the way toward the left of the cell.

The rectangular lattice cell

Why are these the only candidates for isometries or negating isometries in this case? We give the gist of the idea, leaving rigorous proof for the later algebraic section:

If there are to be reflections at all, they must be parallel to the sides of a rectangular cell or along the diagonal of a rhombic cell. If reflections are too close together, they generate a translation smaller than the ones that already appear in the group.

Communications in Visual Mathematics, vol 1, no 1, August 1998.